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19 Jan 2022: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

22 Nov 2022: 11. Let K be a field containing a primitive mth root of unity for some m > 1. ... Show that f and g have the same splittingfield if and only if b = cmar for some c K and r N with gcd(r,m) = 1.

21 Oct 2022: Using this, we complete the proof. Preliminaries. We start by fixing some notation. ... Thus, any simplexappearing in Br(Fn) is contained in some Vi. Since B.

8 Nov 2022: Show that H is normal in G if and only if H is a unionof some conjugacy classes of G. ... 1. |G|gG|Fix(g)|. Deduce that if G acts transitively and |X| > 1, then there is some g G with no xed point.How many distinct ways are there to

24 Nov 2022: 7. Let f: R2 R and (a, b) R2. (a) Suppose that D1f exists and is continuous in some open ball around (a, b), and that. ... b) Suppose instead that D1f exists and is bounded on some open ball around (a, b), and.

13 Jan 2022: b) For some n 2, let X1,. ,Xniid Exp(θ). Find a minimal sufficient statistic T , and compute. ... 4. For some n 2 let X1,. ,Xniid Unif(θ, 2θ) for some θ > 0.

22 Jan 2022: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

28 Feb 2022: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

5 Oct 2022: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... 1.9. Let (E,E,µ) be a measure space. Call a subset N E null if N B for some B E withµ(B) = 0.

12 Oct 2022: We say X is bounded if there is some M R such that for all z X we have ‖z‖ 6 M. ... Show that there. is some subinterval [a,b] of [0, 1] with a < b on which (fn) is uniformly bounded.

22 Jan 2022: 1. 6. Let (An : n N) be a sequence of events in some probability space (,F,P). ... Recall that (1 1/x)x e. as x. ]. Extensions. Some of the following questions have a more puzzle-like quality.

8 Mar 2022: Find the probability densityfunction of the radius of the circle. Have you seen this distribution before for some value of λ? ... TN aNbN. , for some aN,bN. (b) Let αN = bN/3c and βN = b2N/3c.

15 Oct 2022: What happensif Θ = [a,b] for some 0 < a < b <? 9. Let X be multivariate normal N(θ,I), where θ Θ = Rp,p 3, and I is the pp identitymatrix. ... a) Show that the James-Stein estimator θ̃(1) dominates all estimators θ̃(c),c 6= 1.(b) Let θ̂ be the

15 Oct 2022: Suppose that for some θ0 in the interior of Θ and every ε > 0 small enough, wehave S(θ0ε) < 0 < S(θ0ε), and also that Sn has exactly one zero θ̂n ... f(,φ) : φ = Φ(θ) for some θ Θ} equals Φ(θ̂MLE).(b) Now consider a mapping Φ that is

17 Feb 2022: domain. Could there be some other Euclidean function φ making Z[3] into a Euclidean. ... ra sb for some r,s R. Give an example to show that this is not always the case ina UFD.

1 Feb 2022: Further Questions. (12) Is every abelian group the additive group of some ring? ... a) Show that Z$ is a subring of the field Q of rational numbers.(b) Show that any subring R of Q is of the form Z$ for some set $

8 Mar 2022: 2). (c) Show that there is some δ > 0 such that, in any hyperbolic triangle, the union of the δ-neighbourhoods of twoof the sides completely contains the 3rd side. ... Prove that for anyn 2 there is some pair (A,B) and a constellation on (A,B) made of

14 Nov 2022: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... for some λ,µ,ν R (which should turn out in this example to be integers).Does there exist a linear transformation which reduces the pair

1 Feb 2022: Z. Rn. eix eiy2. ||2sd 6 Cn, |x y|2. b) Show that if s = n2 k for some k 2 Z>0, 0 < < 1, then. ... for some 2 R if and only if:. w(x) = cHk1(x1) Hkn(xn)e12 |x|.

12 Aug 2022: Proofs for some results inTopics in Analysis. T. W. Körner. August 12, 2022. ... Now P(z) = 0 if and only if z = ajfor some 1 j m.